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Given the function f(x)=x^(2)+10 x+19, determine the average rate of change of the function over the interval -8 <= x <= -1. Answer:

Given the function f(x)=x2+10x+19 f(x)=x^{2}+10 x+19 , determine the average rate of change of the function over the interval −8≤x≤−1 -8 \leq x \leq-1 .\newlineAnswer:

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Full solution

Q. Given the function f(x)=x2+10x+19 f(x)=x^{2}+10 x+19 , determine the average rate of change of the function over the interval −8≤x≤−1 -8 \leq x \leq-1 .\newlineAnswer:
  1. Calculate f(−8)f(-8): To find the average rate of change of the function f(x)f(x) over the interval [−8,−1][-8, -1], we will use the formula for the average rate of change, which is (f(b)−f(a))/(b−a)(f(b) - f(a)) / (b - a), where aa and bb are the endpoints of the interval.
  2. Calculate f(−1)f(-1): First, we need to calculate the value of the function at the beginning of the interval, which is f(−8)f(-8). We substitute x=−8x = -8 into the function f(x)=x2+10x+19f(x) = x^2 + 10x + 19.\newlinef(−8)=(−8)2+10(−8)+19=64−80+19=3f(-8) = (-8)^2 + 10(-8) + 19 = 64 - 80 + 19 = 3.
  3. Find Average Rate of Change: Next, we calculate the value of the function at the end of the interval, which is f(−1)f(-1). We substitute x=−1x = -1 into the function f(x)=x2+10x+19f(x) = x^2 + 10x + 19.\newlinef(−1)=(−1)2+10(−1)+19=1−10+19=10f(-1) = (-1)^2 + 10(-1) + 19 = 1 - 10 + 19 = 10.
  4. Calculate Average Rate of Change: Now we have the values f(−8)=3f(-8) = 3 and f(−1)=10f(-1) = 10. We can use these to find the average rate of change over the interval [−8,−1][-8, -1].\newlineAverage rate of change = f(−1)−f(−8)−1−(−8)=10−3−1+8=77=1\frac{f(-1) - f(-8)}{-1 - (-8)} = \frac{10 - 3}{-1 + 8} = \frac{7}{7} = 1.
  5. Conclusion: The average rate of change of the function f(x)f(x) over the interval [−8,−1][-8, -1] is 11.

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